We investigate radiative corrections to K → 3π decays. In particular, we extend the non-relativistic framework developed recently to include real and virtual photons and show that, in a well-defined power counting scheme, the results reproduce corrections obtained in the relativistic calculation. Real photons are included exactly, beyond the soft-photon approximation, and we compare the result with the latter. The singularities generated by pionium near threshold are investigated, and a region is identified where standard perturbation theory in the fine structure constant α may be applied. We expect that the formulae provided allow one to extract S-wave ππ scattering lengths from the cusp effect in these decays with high precision.

We investigate two specific Green functions in the framework of chiral perturbation theory. We show that, using analyticity and unitarity, their leading logarithmic singularities can be evaluated in the chiral limit to any desired order in the chiral expansion, with a modest calculational cost. The claim is illustrated with an evaluation of the leading logarithm for the scalar two-point function to five-loop order.Key words: Chiral symmetries, Chiral logarithms PACS: 11.30.Rd, 11.55.Bq Chiral logarithms in the chiral limitIn this article, we discuss the structure of chiral logarithms in the effective low-energy theory of QCD, chiral perturbation theory [1,2]. To simplify the discussion, we consider the case of two flavours u and d. The relevant effective Lagrangian is explicitly known at next-to-next-to-leading order in the chiral expansion [2][3][4]. Its structure iswhere the indices stand for the chiral order of the Lagrangian. We count L n a quantity of order p

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We argue that the linear sigma model at small external momenta is an effective theory for the leading logarithms of chiral perturbation theory. Based on this assumption an attempt is made to sum these leading logarithms using the standard renormalization group techniques, which are valid in renormalizable quantum field theories.

In spite of significant recent progress on the non-perturbative formulation of chiral gauge theories there remained several unsolved problems. One of them is the puzzle that the left-and right-handed projectors, and so the left-and right-handed actions, break CP -symmetry on the lattice. We show in this letter that they break T -symmetry also, while CP T remains intact.